A164927 Sum of the odd prime divisors of numbers with all odd prime divisors of the form 4k+1.

5, 5, 13, 17, 5, 5, 13, 29, 17, 37, 5, 41, 5, 13, 53, 29, 61, 18, 17, 73, 37, 5, 41, 22, 89, 97, 5, 101, 13, 53, 109, 113, 29, 61, 5, 18, 17, 137, 34, 73, 37, 149, 157, 5, 41, 13, 22, 173, 89, 181, 42, 193, 97, 197, 5, 101, 46, 13, 53, 109, 30, 113, 229, 29, 233, 241, 61, 5, 257

Offset: 1

Author

Jonathan Vos Post, Aug 31 2009

Keywords

• easy
• nonn

Comments

We define a sequence b(n) = 5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, ... to consist of those numbers where all odd prime factors are primes contained in A002144, and which have at least one prime factor in this class.

b(n) differs from A009003 which also contains numbers like 30=2*3*5 or 39=3*13, 3 not being in A002144.

b(n) essentially contains elements of A004613 multiplied by powers of 2.

a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.

Sum of distinct Pythagorean prime divisors of integers whose only odd prime divisors are Pythagorean primes A002144.

Analogous sequence for primes of form 4k+3 is A164928.

Analogous sequence for primes of form 6k+1 is A164929.

Analogous sequence for primes of form 6k+5 is A164930.

Examples

			a(18) = 18 because b(18) = 65 = 5*13, and 5+13 = 18.
The smallest number, all of whose prime factors are of form 4n+1, whose sum of distinct prime factors is prime: 1885 = 5 * 13 * 29; and 5 + 13 + 29 = 47.
		

Crossrefs

Cf. A000040, A002144, A009003, A164927-A164930.

Programs

Maple

isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 4 <> 1 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164927 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164927(n),n=1..120) ; # R. J. Mathar, Sep 09 2009

Extensions

Edited, definition clarified by R. J. Mathar, Sep 08 2009